Exercise 7, § 9

We retain the notations of the proof of Th. 2.

a) Show that

b) Take $(\cdot | \cdot )$ to be the canonical bilinear form $\Phi_R$ (Chap. VI, §1, no. 12). Show that

$^{18}$For more details on this exercise, see: H. FREUDENTHAL, Zur Berechnung der

Charaktere der halbeinfachen Lieschen Gruppen, Proc. Kon. Akad. Wet. Amsterdam, Vol. LVII (1954), pp. 369-376.

$f_{\rho}(J(e^\mu))\equiv d_\mu T^N(1 +\frac{T^2}{48}(\mu|\mu))$ (mod$.T^{N+3}\mathbf{R}[[T]]$),

where $d_\mu=\prod_{\alpha\in R_+}(\mu|\alpha )$.

c) From b) and the equality $J(e^{\lambda+\rho}) =$ ch(E)$.J(e^{\rho})$, deduce the formula

$\sum(\mu|\rho )^2$ dim $E^\mu=$ dim E24 $(\lambda |\lambda + 2\rho )$.

$\mu\in P$

d) Assume that $\mathfrak{g}$ is simple. Show that $(\rho |\rho ) =$ dim $\mathfrak{g}/24$. (Apply $c$) with $\lambda$ the highest root of R, and use Exerc. 3 of §6.)